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Mirrors > Home > MPE Home > Th. List > Mathboxes > lsatcvat | Structured version Visualization version GIF version |
Description: A nonzero subspace less than the sum of two atoms is an atom. (atcvati 30480 analog.) (Contributed by NM, 10-Jan-2015.) |
Ref | Expression |
---|---|
lsatcvat.o | ⊢ 0 = (0g‘𝑊) |
lsatcvat.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
lsatcvat.p | ⊢ ⊕ = (LSSum‘𝑊) |
lsatcvat.a | ⊢ 𝐴 = (LSAtoms‘𝑊) |
lsatcvat.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
lsatcvat.u | ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
lsatcvat.q | ⊢ (𝜑 → 𝑄 ∈ 𝐴) |
lsatcvat.r | ⊢ (𝜑 → 𝑅 ∈ 𝐴) |
lsatcvat.n | ⊢ (𝜑 → 𝑈 ≠ { 0 }) |
lsatcvat.l | ⊢ (𝜑 → 𝑈 ⊊ (𝑄 ⊕ 𝑅)) |
Ref | Expression |
---|---|
lsatcvat | ⊢ (𝜑 → 𝑈 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsatcvat.o | . . 3 ⊢ 0 = (0g‘𝑊) | |
2 | lsatcvat.s | . . 3 ⊢ 𝑆 = (LSubSp‘𝑊) | |
3 | lsatcvat.p | . . 3 ⊢ ⊕ = (LSSum‘𝑊) | |
4 | lsatcvat.a | . . 3 ⊢ 𝐴 = (LSAtoms‘𝑊) | |
5 | lsatcvat.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
6 | 5 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑊 ∈ LVec) |
7 | lsatcvat.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
8 | 7 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑈 ∈ 𝑆) |
9 | lsatcvat.q | . . . 4 ⊢ (𝜑 → 𝑄 ∈ 𝐴) | |
10 | 9 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑄 ∈ 𝐴) |
11 | lsatcvat.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝐴) | |
12 | 11 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑅 ∈ 𝐴) |
13 | lsatcvat.n | . . . 4 ⊢ (𝜑 → 𝑈 ≠ { 0 }) | |
14 | 13 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑈 ≠ { 0 }) |
15 | lsatcvat.l | . . . 4 ⊢ (𝜑 → 𝑈 ⊊ (𝑄 ⊕ 𝑅)) | |
16 | 15 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑈 ⊊ (𝑄 ⊕ 𝑅)) |
17 | simpr 488 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → ¬ 𝑄 ⊆ 𝑈) | |
18 | 1, 2, 3, 4, 6, 8, 10, 12, 14, 16, 17 | lsatcvatlem 36813 | . 2 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑈 ∈ 𝐴) |
19 | 5 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑊 ∈ LVec) |
20 | 7 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑈 ∈ 𝑆) |
21 | 11 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑅 ∈ 𝐴) |
22 | 9 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑄 ∈ 𝐴) |
23 | 13 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑈 ≠ { 0 }) |
24 | lveclmod 20156 | . . . . . . . . 9 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
25 | 5, 24 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ LMod) |
26 | lmodabl 19959 | . . . . . . . 8 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel) | |
27 | 25, 26 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Abel) |
28 | 2 | lsssssubg 20008 | . . . . . . . . 9 ⊢ (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊)) |
29 | 25, 28 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝑊)) |
30 | 2, 4, 25, 9 | lsatlssel 36761 | . . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ 𝑆) |
31 | 29, 30 | sseldd 3911 | . . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ (SubGrp‘𝑊)) |
32 | 2, 4, 25, 11 | lsatlssel 36761 | . . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ 𝑆) |
33 | 29, 32 | sseldd 3911 | . . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ (SubGrp‘𝑊)) |
34 | 3 | lsmcom 19256 | . . . . . . 7 ⊢ ((𝑊 ∈ Abel ∧ 𝑄 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊)) → (𝑄 ⊕ 𝑅) = (𝑅 ⊕ 𝑄)) |
35 | 27, 31, 33, 34 | syl3anc 1373 | . . . . . 6 ⊢ (𝜑 → (𝑄 ⊕ 𝑅) = (𝑅 ⊕ 𝑄)) |
36 | 35 | psseq2d 4017 | . . . . 5 ⊢ (𝜑 → (𝑈 ⊊ (𝑄 ⊕ 𝑅) ↔ 𝑈 ⊊ (𝑅 ⊕ 𝑄))) |
37 | 15, 36 | mpbid 235 | . . . 4 ⊢ (𝜑 → 𝑈 ⊊ (𝑅 ⊕ 𝑄)) |
38 | 37 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑈 ⊊ (𝑅 ⊕ 𝑄)) |
39 | simpr 488 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → ¬ 𝑅 ⊆ 𝑈) | |
40 | 1, 2, 3, 4, 19, 20, 21, 22, 23, 38, 39 | lsatcvatlem 36813 | . 2 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑈 ∈ 𝐴) |
41 | 29, 7 | sseldd 3911 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊)) |
42 | 3 | lsmlub 19067 | . . . . . . 7 ⊢ ((𝑄 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊) ∧ 𝑈 ∈ (SubGrp‘𝑊)) → ((𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈) ↔ (𝑄 ⊕ 𝑅) ⊆ 𝑈)) |
43 | 31, 33, 41, 42 | syl3anc 1373 | . . . . . 6 ⊢ (𝜑 → ((𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈) ↔ (𝑄 ⊕ 𝑅) ⊆ 𝑈)) |
44 | ssnpss 4027 | . . . . . 6 ⊢ ((𝑄 ⊕ 𝑅) ⊆ 𝑈 → ¬ 𝑈 ⊊ (𝑄 ⊕ 𝑅)) | |
45 | 43, 44 | syl6bi 256 | . . . . 5 ⊢ (𝜑 → ((𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈) → ¬ 𝑈 ⊊ (𝑄 ⊕ 𝑅))) |
46 | 45 | con2d 136 | . . . 4 ⊢ (𝜑 → (𝑈 ⊊ (𝑄 ⊕ 𝑅) → ¬ (𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈))) |
47 | ianor 982 | . . . 4 ⊢ (¬ (𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈) ↔ (¬ 𝑄 ⊆ 𝑈 ∨ ¬ 𝑅 ⊆ 𝑈)) | |
48 | 46, 47 | syl6ib 254 | . . 3 ⊢ (𝜑 → (𝑈 ⊊ (𝑄 ⊕ 𝑅) → (¬ 𝑄 ⊆ 𝑈 ∨ ¬ 𝑅 ⊆ 𝑈))) |
49 | 15, 48 | mpd 15 | . 2 ⊢ (𝜑 → (¬ 𝑄 ⊆ 𝑈 ∨ ¬ 𝑅 ⊆ 𝑈)) |
50 | 18, 40, 49 | mpjaodan 959 | 1 ⊢ (𝜑 → 𝑈 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 847 = wceq 1543 ∈ wcel 2111 ≠ wne 2941 ⊆ wss 3875 ⊊ wpss 3876 {csn 4550 ‘cfv 6389 (class class class)co 7222 0gc0g 16957 SubGrpcsubg 18550 LSSumclsm 19036 Abelcabl 19184 LModclmod 19912 LSubSpclss 19981 LVecclvec 20152 LSAtomsclsa 36738 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-rep 5188 ax-sep 5201 ax-nul 5208 ax-pow 5267 ax-pr 5331 ax-un 7532 ax-cnex 10798 ax-resscn 10799 ax-1cn 10800 ax-icn 10801 ax-addcl 10802 ax-addrcl 10803 ax-mulcl 10804 ax-mulrcl 10805 ax-mulcom 10806 ax-addass 10807 ax-mulass 10808 ax-distr 10809 ax-i2m1 10810 ax-1ne0 10811 ax-1rid 10812 ax-rnegex 10813 ax-rrecex 10814 ax-cnre 10815 ax-pre-lttri 10816 ax-pre-lttrn 10817 ax-pre-ltadd 10818 ax-pre-mulgt0 10819 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rmo 3070 df-rab 3071 df-v 3417 df-sbc 3704 df-csb 3821 df-dif 3878 df-un 3880 df-in 3882 df-ss 3892 df-pss 3894 df-nul 4247 df-if 4449 df-pw 4524 df-sn 4551 df-pr 4553 df-tp 4555 df-op 4557 df-uni 4829 df-int 4869 df-iun 4915 df-iin 4916 df-br 5063 df-opab 5125 df-mpt 5145 df-tr 5171 df-id 5464 df-eprel 5469 df-po 5477 df-so 5478 df-fr 5518 df-we 5520 df-xp 5566 df-rel 5567 df-cnv 5568 df-co 5569 df-dm 5570 df-rn 5571 df-res 5572 df-ima 5573 df-pred 6169 df-ord 6225 df-on 6226 df-lim 6227 df-suc 6228 df-iota 6347 df-fun 6391 df-fn 6392 df-f 6393 df-f1 6394 df-fo 6395 df-f1o 6396 df-fv 6397 df-riota 7179 df-ov 7225 df-oprab 7226 df-mpo 7227 df-om 7654 df-1st 7770 df-2nd 7771 df-tpos 7977 df-wrecs 8056 df-recs 8117 df-rdg 8155 df-1o 8211 df-er 8400 df-en 8636 df-dom 8637 df-sdom 8638 df-fin 8639 df-pnf 10882 df-mnf 10883 df-xr 10884 df-ltxr 10885 df-le 10886 df-sub 11077 df-neg 11078 df-nn 11844 df-2 11906 df-3 11907 df-sets 16730 df-slot 16748 df-ndx 16758 df-base 16774 df-ress 16798 df-plusg 16828 df-mulr 16829 df-0g 16959 df-mre 17102 df-mrc 17103 df-acs 17105 df-mgm 18127 df-sgrp 18176 df-mnd 18187 df-submnd 18232 df-grp 18381 df-minusg 18382 df-sbg 18383 df-subg 18553 df-cntz 18724 df-oppg 18751 df-lsm 19038 df-cmn 19185 df-abl 19186 df-mgp 19518 df-ur 19530 df-ring 19577 df-oppr 19654 df-dvdsr 19672 df-unit 19673 df-invr 19703 df-drng 19782 df-lmod 19914 df-lss 19982 df-lsp 20022 df-lvec 20153 df-lsatoms 36740 df-lcv 36783 |
This theorem is referenced by: lsatcvat2 36815 |
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